Symmetry protected phases in inhomogeneous spin chains

It has been shown recently that inhomogenous spin chains can exhibit exotic phenomena such as the breaking of the area law of the entanglement entropy. An example is given by the rainbow model where the exchange coupling constants decrease from the center of the chain. Here we show that by folding the chain around its center, the long-range entanglement becomes short-range which can lead to topological phases protected by symmetries (SPT). The phases are trivial for bond-centered foldings, and non trivial for site-centered ones. In the latter case, the folded spin 1/2 chain with $U(1)$ symmetry belongs to the Su-Schrieffer-Heeger class, while the folded chain with $SU(2)$ symmetry is in the Haldane phase. Finally, we extend these results to higher spin chains where we find a correspondence between the symmetry protection of gapped and gapless phases.